Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 37-46: (a) Sketch the graph of \(f\). (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing, is decreasing, or is constant. $$ f(x)=x^{2}-1 $$

Step-by-Step Solution

Answer

Domain: \( (-\infty, \infty) \). Range: \( [-1, \infty) \). f is increasing on \((0,\infty)\) and decreasing on \((-\infty,0)\).

Step 1: Sketch the Graph of f

The function given is \( f(x) = x^2 - 1 \). This is a quadratic function, which means its graph will be a parabola. Since it is in the form \( ax^2 + bx + c \) where \( a = 1 \), the parabola opens upwards. The graph intersects the y-axis at \( (0, -1) \) because this is the constant term. Vertex can be found at \( x = -\frac{b}{2a} = 0 \), so the vertex is \( (0, -1) \). By plotting these points and a few others, you can draw a U-shaped curve extending upwards from the vertex.

Step 2: Find the Domain and Range

The function \( f(x) = x^2 - 1 \) is a polynomial. Polynomials are defined for all real numbers, so the domain \( D \) is \( (-\infty, \infty) \). For the range, the vertex \( (0, -1) \) is the minimum point of the parabola because it opens upwards. Thus, the smallest value \( f(x) \) can take is \(-1\), and it increases without bound. Hence, the range \( R \) is \( [-1, \infty) \).

Step 3: Determine Intervals of Increasing and Decreasing

To find where the function is increasing or decreasing, we look at the derivative \( f'(x) = 2x \). The derivative is zero at \( x = 0 \), which is the vertex of the parabola. For \( x < 0 \), \( f'(x) < 0 \), and for \( x > 0 \), \( f'(x) > 0 \). This indicates that the function is decreasing on \((-fty, 0)\) and increasing on \((0, fty)\). The function is not constant anywhere in this interval.

Key Concepts

Parabola

A parabola is a U-shaped curve that can open either upwards or downwards. For the quadratic function \( f(x) = x^2 - 1 \), the parabola opens upwards because the coefficient of the \( x^2 \) term is positive. The general structure of a quadratic function is \( ax^2 + bx + c \), where 'a' determines the direction of the parabola. Here, \( a = 1 \), \( b = 0 \), and \( c = -1 \).

• The vertex of the parabola is the point where it changes direction, acting as the minimum or maximum point.
• For \( f(x) = x^2 - 1 \), the vertex is at \( (0, -1) \).
• The parabola intersects the y-axis at the point \( (0, -1) \), as \( c = -1 \).

The graph of this function is symmetric about the y-axis, meaning the left and right sides are mirror images.

Domain and Range

Understanding the domain and range of a function is crucial in graph interpretation. For the quadratic \( f(x) = x^2 - 1 \), the domain is all real numbers, represented as \( (-\infty, \infty) \). This is because there are no restrictions on the values of \( x \) you can use in a quadratic.

• The domain indicates all the input values \( x \) for which the function is defined.
• For polynomials like \( f(x) = x^2 - 1 \), you can plug in any real number and get a valid output.

The range deals with all possible output values \( f(x) \). Since the vertex of the parabola is at \( (0, -1) \) and the parabola opens upwards, the minimum value the function can take is \( -1 \). This results in the range being \( [-1, \infty) \), meaning it includes all real numbers from \( -1 \) upwards.

Increasing and Decreasing Intervals

Identifying intervals where a function is increasing or decreasing helps understand its behavior. To determine this for \( f(x) = x^2 - 1 \), we can use its derivative \( f'(x) = 2x \). The derivative provides the slope at any point \( x \):

• When \( f'(x) > 0 \), the function is increasing.
• When \( f'(x) < 0 \), the function is decreasing.

At the vertex \( (0, -1) \), the derivative is zero, indicating neither increase nor decrease.- For \( x < 0 \), \( f'(x) = 2x < 0 \), indicating that the function is decreasing.- For \( x > 0 \), \( f'(x) = 2x > 0 \), indicating that the function is increasing.Therefore, \( f(x) \) is decreasing in the interval \((-\infty, 0)\) and increasing in the interval \((0, \infty)\). This information reflects the overall shape and direction of the parabola around its vertex.